Riesz-Jacobi transforms as principal value integrals

TL;DR

This paper provides a detailed integral representation of Riesz-Jacobi transforms, distinguishing between odd and even orders, and suggests that similar existing results in related areas need refinement and correction.

Contribution

It establishes principal value integral representations for Riesz-Jacobi transforms and clarifies their singularity properties, challenging previous assumptions in related settings.

Findings

Odd order transforms are principal value integrals with non-integrable singularities.

Even order transforms are regular and include the identity operator.

Results suggest a need to revisit similar existing theorems in related orthogonal expansions.

Abstract

We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz-Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz-Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.